Answer:
[tex]\mathbf{r = (-5.064 \ \hat i + 6.536 \ \hat j) km}[/tex]
Explanation:
Given that:
A cyclist rides 4.0 km due west, then 12.0 km 33° west of north. From this point she rides 9.0 km due east.
Let:
[tex]r_1 = 4.0 \ km \ due \ west \\ \\ r_2 = 12.0 \ km \ \ \ \ \ \theta = 33^0 \ west \ of \ north \\ \\ r_3 = 9.0 \ km \ due \ east[/tex]
Assuming that:
east is the + x axis and has a unit vector of [tex]\hat i[/tex]
north is the +y axis and has a unit vector of [tex]\hat j[/tex]
west is the - x axis and has a unit vector of [tex]-\hat {i}[/tex]
south is the - y axis and has a unit vector of -[tex]\hat j[/tex]
The displacement [tex]r_x[/tex] in a given direction of x can be expressed by the formula:
[tex]r_x = \sum r_i[/tex]
[tex]r_x = -4-12 cos (33)+9[/tex]
[tex]r_x = - 5.064 \ km[/tex]
The displacement [tex]r_y[/tex] in a given direction of y can be expressed by the formula:
[tex]r_y = \sum r_j[/tex]
[tex]r_y = 12 \ \ * s in (33)[/tex]
[tex]r_y = 6.536 \ km[/tex]
The final displacement can be expressed by the relation;
[tex]r = r_x \hat i + r_y \hat j[/tex]
[tex]\mathbf{r = (-5.064 \ \hat i + 6.536 \ \hat j) km}[/tex]
